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Week 3: Thermal History of the Universe

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Week 3: Thermal History of the Universe Week 3: Thermal History of the Universe Cosmology, Ay127, Spring 2008 April 21, 2008 1 Brief Overview Before moving on, let’s review some of the high points in the history of the Universe: T 10−4 eV, t 1010 yr: today; baryons and the CMB are entirely decoupled, stars and • ∼ ∼ galaxies have been around a long time, and clusters of galaxies (gravitationally bound systems of 1000s of galaxies) are becoming common. ∼ T 10−3 eV, t 109 yr; the first stars and (small) galaxies begin to form. • ∼ ∼ T 10−1−2 eV, t millions of years: baryon drag ends; at earlier times, the baryons are still • ∼ ∼ coupled to the CMB photons, and so perturbations in the baryon cannot grow; before this time, the gas distribution in the Universe remains smooth. T eV, t 400, 000 yr: electrons and protons combine to form hydrogen atoms, and CMB • ∼ ∼ photons decouple; at earlier times, photons are tightly coupled to the baryon fluid through rapid Thomson scattering from free electrons. T 3 eV, t 10−(4−5) yrs: matter-radiation equality; at earlier times, the energy den- • ∼ ∼ sity of the Universe is dominated by radiation. Nothing really spectacular happens at this time, although perturbations in the dark-matter density can begin to grow, providing seed gravitational-potential wells for what will later become the dark-matter halos that house galaxies and clusters of galaxies. T keV, t 105 sec; photons fall out of chemical equilibrium. At earlier times, interac- • ∼ ∼ tions that can change the photon number occur rapidly compared with the expansion rate; afterwards, these reactions freeze out and CMB photons are neither created nor destroyed. T 10 0.1 MeV, t seconds-minutes. Big-bang nucleosynthesis (BBN). Neutrons and • ∼ − ∼ protons first combine to form D, 4He, 3He, and 7Li nuclei. Quite remarkably, the theory for this is very well developed and agrees very impressively with a variety of observations. T 100 300 MeV, t 10−5 sec. The quark-hadron phase transition. This is when quarks • ∼ − ∼ and gluons first become bound into neutrons and protons. We are pretty sure that this must 1 have happened, but the details are not well understood, and there are still no signatures of this phase transition that have been observed. This is also when axions are produced, if they exist and are the dark matter. T 10s 100s GeV, t 10−8 sec. If dark matter is composed of supersymmetric particles, or • ∼ − ∼ some other weakly interacting massive particle (WIMP), then this is when their interactions freeze out and their cosmological abundance fixed. This is speculative. T 1012 GeV, t 10−30 sec. The Peccei-Quinn phase transition, if the Peccei-Quinn • ∼ ∼ mechanism is the correct explanation for the strong-CP problem. This is highly speculative. T 1016 GeV, t 10−38 sec. This is when the GUT (grand-unified theory) phase transition • ∼ ∼ occurs. At earlier times, the strong and electroweak interactions are indistinguishable. This is highly speculative. T 1019 GeV, t 10−43 sec. Strings? quantum gravity? quantum birth of the Universe? • ∼ ∼ This is all highly speculative. Basically, at these temperatures, the energy densities are so high that the classical treatment of general relativity is no longer reliable. Such early times can only be understood with a quantum theory of gravity, which we don’t have. 2 Thermal History of the Universe Thermal equilibrium of particle species in the early Universe is maintained by scattering of parti- cles. Thus, equilibrium is maintained (and the expansion of the Universe can be considered to be adiabatic) as long as the interaction rate Γ = nσv is greater than the expansion rate, H T 2/m . ∼ Pl Here, n is the number density of (other) particles that the particle in question can scatter from, and σ is the cross section for the scattering process in question, and v the relative velocity between the reacting particles. At this point, it is important to distinguish between statistical equilibrium and chemical equilibrium. In chemical equilibrium, the reactions that create and destroy the particles in question occur faster than the expansion, and so the chemical potential retains its equilibrium value (as determined by, e.g., µ + µ = µ + µ for reactions i + j k + l). In statistical equilibrium, the i j k l ↔ elastic scattering reactions that maintain the thermal energy distribution of particles occur faster than the expansion. It is possible for a particular particle species to be in statistical equilibrium, but not in chemical equilibrium. I think you may be able to prove that if chemical equilibrium is maintained, then so is statistical equilibrium, but I’m not entirely sure. True thermal equilibrium occurs when the particle species in question is in statistical and chemical equilibrium. Suppose now that the interaction rate Γ scales with temperature as Γ T n. Then, the number of ∝ interactions the particle will undergo after some time t is ∞ Γ(T ) T T ′ n dT ′ Γ 1 N = Γ(t′) dt′ = T 2 = . (1) int 2H(T ) T T ′3 H n 2 Zt Z0 t − Therefore, for n> 2 particles interact < 1 time after the time when Γ H. This is simply because ≃ during radiation domination, H T 2,∼ so if n> 2, the scattering rate Γ falls below H after Γ = H, ∝ and the interactions of the particles are said to freeze out. 2 Let’s start with neutrino decoupling. At temperatures T > m 0.5 MeV, electrons and positrons e ≃ are abundant in the Universe, and chemical equilibrium of∼ neutrinos is maintained by νν¯ e+e− ↔ reactions through a virtual Z0 boson and additionally (for the electron neutrino) through W ± exchange, and statistical equilibrium is additionally maintained by neutrino-electron elastic scattering, which occurs through W ± exchange (for electron neutrinos) and Z0 exchange (for all three neutrinos). The cross section for these processes can be calculated using standard weak-interaction theory, which is beyond the scope of this course. The result for the cross sections turns out to be σ 2 4 2 −2 ∼ (α /mW )Eν , where α 10 is the fine-structure constant, mW,Z 100 GeV is the gauge-boson ∼ 2 ∼ mass, and Eν the neutrino energy. The α factor arises because there are two vertices in the matrix element (the Feynman diagram), and the cross section is proportional to the matrix element −4 2 squared. The mW,Z arises because of a factor mW,Z appears in the propagator for the virtual gauge 2 −2 boson that is exchanged. We then add the factor Eν to get the right dimensions [(energy) ] for the cross section. Noting that the abundance of electrons (when they are still relativistic is n T 3, ∼ that neutrinos travel at velocities v c, and that the typical neutrino energy is E T , the ≃ ν ∼ interaction rate for neutrino conversion to electrons and for neutrino-electron elastic scattering is Γ α2T 5/m4 , and the ratio to the expansion rate is (Γ /H) α2T 3m /m4 (T/MeV)3. int ∼ W,Z int ∼ Pl W,Z ∼ Therefore, for temperatures T >MeV, neutrinos are in thermal equilibrium, and they decouple at temperatures T MeV. Of course,∼ all this has been pretty fast and loose, but it turns out that ∼ when you put all the factors of π, 2, etc. in the right place, you get this result. We then note that these neutrinos then free stream throughout the remaining history of the Universe, and their temperature today is (4/11)1/3 the photon temperature because, as discussed last week, the photon number increases by (11/4) during electron-positron annihilation. 3 Recombination In the early Universe, at temperatures T eV, the atoms in the Universe are ionized, and photons ≫ are tightly coupled to the baryons through Thomson scattering from the electrons. At a temperature T eV, electrons and protons combine to form hydrogen atoms, free electrons disappear, and ∼ without any electrons to scatter from, photons decouple and free stream through the Universe. These are the CMB photons we see today. The temperature at which recombination occurs can be determined roughly from the Saha equation. Let’s ignore the helium in the Universe. Then, there can be electrons, protons, and bound hydrogen 3 atoms, with abundances ne, np, and nH , and the baryon density is nB = np + nH. In terms of the temperature and chemical potentials, the abundances are m T 3/2 µ m n = g i exp i − i . (2) i i 2π T From the reaction e− + p H, we have µ + µ = µ , which allows us to write, ↔ p e H m T 3/2 µ + µ m g m T −3/2 n = g H exp p e − H = H n n e eB/T , ] (3) H h 2π T g g p e 2π p e where the binding energy B m + m m = 13.6 eV, and we have used m m in the ≡ p e − H p ≃ H prefactor. Defining an ionization fraction X n /n , and g = g = 2 and g = 4, and e ≡ p H p e H n = ηn , where η = 5.1 10−10(Ω h2/0.019) is the baryon-to-photon ratio, and n = n and B γ × b p e n = (1 X )n , we find (this equation needs to be checked!) H − e p eq −3/2 (1 Xe ) 4√2ζ(3) T − = η eB/T . (4) (Xeq)2 √π m e e This is known as the Saha equation. Keep in mind that here that T (z) = T0(1 + z). At high temperatures, one gets an ionization fraction X 1, and X 0 at low temperatures.
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